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In Mathematics , the binomial theorem is an important Formula giving the expansion of Powers of Sum s. Its simplest version reads

:(x+y)^n=\sum_{k=0}^n{n \choose k}x^ky^{n-k}\quad\quad\quad(1)

whenever ''n'' is any non-negative integer, the numbers

:{n \choose k}= rac{n!}{k!(n-k)!}

are the Binomial Coefficient s, and n! denotes the Factorial of ''n''.

This formula, and the Triangular Arrangement Of The Binomial Coefficients , are often attributed to Blaise Pascal who described them in the 17th Century . It was, however, known to the Chinese Mathematician Yang Hui in the 13th Century , the earlier Persian Mathematician Omar Khayyám in the 11th Century , and the even earlier Indian Mathematician Pingala in the 3rd Century BC .

For example, here are the cases ''n'' = 2, ''n'' = 3 and ''n'' = 4:
:(x + y)^2 = x^2 + 2xy + y^2\,
:(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\,
:(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4.\,

Formula (1) is valid for all Real or Complex numbers ''x'' and ''y'', and more generally for any elements ''x'' and ''y'' of a Semiring as long as ''xy'' = ''yx''.


NEWTON'S GENERALIZED BINOMIAL THEOREM

Isaac Newton generalized the formula to other exponents by considering an Infinite Series :

:{(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^k y^{r-k}\quad\quad\quad(2)}

where ''r'' can be any Complex Number (in particular ''r'' can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by

:{r \choose k}={1 \over k!}\prod_{n=0}^{k-1}(r-n)= rac{r(r-1)(r-2)\cdots(r-(k-1))}{k!}\,

In case ''k'' = 0, this is a Product Of No Numbers At All and therefore equal to 1, and in case ''k'' = 1 it is equal to ''r'', as the additional factors (''r'' − 1), etc., do not appear.

Another way to express this quantity is

:{r \choose k}= rac{(-1)^k}{k!}(-r)_k,

which is important when one is working with infinite series and would like to represent them in terms of Generalized Hypergeometric Function s. The notation (\cdot)_k is the Pochhammer Symbol . This form is vital in applied mathematics, for example, when evaluating the formulas that model the statistical properties of the phase-front curvature of a light wave as it propagates through optical atmospheric turbulence.

A particularly handy but non-obvious form holds for the reciprocal power:
: rac{1}{(1-x)^r}=\sum_{k=0}^\infty {r+k-1 \choose k} x^k \equiv \sum_{k=0}^\infty {r+k-1 \choose r-1} x^k.

For a more extensive account of Newton's generalized binomial theorem, see Binomial Series .

  Formula (2) Is Also Valid For Elements ''x'' And ''y'' Of A "http://wwwinformationdelightinfo/encyclopedia/entry/Banach_algebra" class="copylinks">Banach Algebra as long as ''xy''&nbsp=&nbsp''yx'', ''y'' is invertible and ''x/y'' < 1